## Mathematical Recreations

Yes, this is the classic airplane on a treadmill problem. I try to steer clear of problems that have been thoroughly adressed elsewhere, but this problem is special. It has been addressed so poorly elsewhere that I'm willing to try to provide a solid (and correct) discussion. Although this is more of a physics topic than a mathematical topic, it has the same flavor as other topics at MathRec. I won't be spoiling anything by revealing that the solution depends on carefully interpreting the question. In fact, I'm presenting it as two problems which are not equivalent:

A plane is standing on a runway that can move (some sort of band conveyer). The plane moves in one direction, while the conveyer moves in the opposite direction. This conveyer has a control system that tracks the plane's speed and tunes the speed of the conveyer to be exactly the same (but in the opposite direction). Can the plane take off?

and

A plane is standing on a runway that can move (some sort of band conveyer). The conveyer belt exactly matches the speed of the wheels at any given time, moving in the opposite direction of rotation. Can the plane take off?

The answer does not depend on limitations in the aircraft parts. Specifically, you can assume that such a conveyer belt actually exists; the wheels turn on their axles without friction; the wheels do not slip on the treadmill; and the wheels do not melt or explode due to rolling friction on the conveyer belt. Other than having idealized wheels, there is nothing special about the airplane. (The conveyer belt is special, but doesn't violate any laws of physics.)